A function f is defined as f ( x ) = { c o s x − 1 x 2 f o r x ≠ 0 − 1 2 f o r x = 0 Using the first three non zero terms of the Taylor series for cosx about c o s x, write the first three non zero terms of the Taylor series for f about c o s x.

Question

A function   f is defined as  f  (  x  )  =      {                                                                              c                o                s                x                −                1                                            x                                  2                                                                              f          o          r                    x          ≠          0                                                                                        −                1                            2                                                f          o          r                    x          =          0                        Using the first three non zero terms of the Taylor series for cosx about   c  o  s    x, write the first three non zero terms of the Taylor series for   f about   c  o  s    x.

Lintynx · Accepted Answer

For all x we have  cos  ⁡  x  −  1  =  −      1    2        x    2    +      1          4      !            x    4    −      1          6      !            x    6    +  …hence in a neighbourhood of   0 (and   x  ≠  0)In particular, it is not true that       f    ′    (  0  )  ,      f    ″    (  0  )  ,      f    ‴    (  0  )  ,  … are all zero. Only odd derivatives vanish (that&#039;s because   f is an even function).

A function f is defined as f(x)={((cosx−1)/x^2)forx≠0,((−1)/2) for x=0:} Using the first three non zero terms of the Taylor series for cosx about x=0, write the first three non zero terms of the Taylor series for f about x=0.

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